The sea-ice floe size distribution (FSD) characterizes
the sea-ice response to atmospheric and oceanic forcing and is important for
understanding and modeling the evolving ice pack in a warming Arctic. FSDs
are evaluated from 78 floe-segmented high-resolution (1 m) optical satellite
images capturing a range of settings and sea-ice states during spring
through fall from 1999 to 2014 in the Canada Basin. For any given image, the
structure of the FSD is found to be sensitive to a classification threshold
value (i.e., to specify an image pixel as being either water or ice) used in
image segmentation, and an approach to account for this sensitivity is
presented. The FSDs are found to exhibit a single power-law regime between
floe areas 50 m2 and 5 km2, characterized by exponents (slopes in
log-log space) in the range -2.03 to -1.65. A distinct linear relationship
between slopes and sea-ice concentrations is found, with steeper slopes
(i.e., a larger proportion of smaller to larger floes) corresponding to
lower sea-ice concentrations. Further, a seasonal variation in slopes is
found for fixed sites in the Canada Basin that undergo a seasonal cycle in
sea-ice concentration, while sites with extensive sea-ice cover year-round
do not exhibit any seasonal change in FSD properties. Our results suggest
that sea-ice concentration should be considered in any characterization of a
time-varying FSD (for use in sea-ice models, for example).
Introduction
The Arctic Ocean is covered perennially to varying extents by sea ice
floating in discrete fragments called floes, which range in size from O(1) m
to O(100) km (Untersteiner, 1986). This assortment of sizes, which may be
described by a sea-ice floe size distribution (FSD, see Rothrock and
Thorndike, 1984), influences and is influenced by the ice pack response to
thermal and dynamic atmospheric and oceanic forcing; for example, a
distribution with a larger fraction of smaller, thinner floes will melt more
rapidly (e.g., via lateral melting) (Steele, 1992) and deform and drift
with less resistance than a field comprised of more larger floes. In turn,
the FSD influences energetics and mixing in the upper ocean through a
variety of processes, such as spatially variable momentum transfer and
buoyancy fluxes that generate small-scale ocean flows (e.g., Mensa and
Timmermans, 2017; Smith et al., 2002). Bateson et al. (2020) account for
varying floe sizes in a sea-ice model (developed for use in a climate model)
via an FSD that is iteratively modified by melt/growth and dynamical
processes; they demonstrate that melt patterns (e.g., basal vs. lateral
melt) differ significantly when a size distribution is accounted for (see
also Roach et al., 2018). Accurate observational characterization of the FSD
yields insight into the physics of the ice cover and its surroundings and
provides validation of Arctic modeling studies that incorporate the FSD to
more accurately represent these processes and their seasonality (e.g.,
Horvat and Tziperman, 2015; Roach et al., 2019; Zhang et al., 2015, 2016).
The sea-ice FSD has been characterized extensively in observations since the
seminal paper of Rothrock and Thorndike (1984); the FSD may be quantified in
a number of ways, for example as the number of floes per unit area of the
region in question which have sizes that are not smaller than a given size.
In general, the FSD resembles a single power law (e.g., Gherardi and
Lagomarsino, 2015; Hwang et al., 2017; Stern et al., 2018b) or two distinct
power laws depending on floe scales (Geise et al., 2017; Steer et al., 2008;
Toyota et al., 2011, 2006), although alternate distributions
have been explored (see, e.g., Herman, 2010, and the discussion by Stern et
al., 2018a). There are limited FSD characterizations that span a
comprehensive range of floe scales, from O(1) m to more than O(10) km (see
Stern et al., 2018a). This is in part due to a reliance on high-resolution
aerial photography with limited area coverage and sampling. While Stern et
al. (2018b) find that a single-power law may describe the FSD across floe
scales ranging from 10 m to 30 km, it remains an open question as to whether
a single power law holds across all floe scales and in all settings, or
whether there may be two distinct power-law regimes. The seasonal evolution
of observed FSDs has been the subject of several recent observational
studies (Hwang et al., 2017; Perovich and Jones, 2014; Stern et al., 2018b),
each of which finds a steepening of the FSD slope into summer. This slope
increase in the melt season is thought to be related to the breakup of
floes beginning in the spring in tandem with melt through the summer
reducing the proportion of larger to smaller floes (e.g., Stern et al.,
2018b).
A collection of high-resolution optical satellite images, spanning nearly
two decades, from different locations within the Canada Basin, allows us to
test and refine previous findings for a variety of settings, and for floe
sizes in the range of 5 m2 to 100 km2. In the
next section, we introduce the collection of images and describe our image
segmentation methodology and FSD construction. In Sect. 3, we show how FSDs
exhibit a single power-law behavior spanning the full range of floe sizes
and provide evidence for a shoaling of the slope of the distribution (i.e.,
increased ratio of larger to smaller floes) as sea-ice concentration
increases. This finding is consistent with a seasonal evolution of the FSD
found here, which we describe in context with previous studies in Sect. 3.4.
Results are summarized and discussed in Sect. 4.
Data and methodsSatellite imagery and environmental parameters
We perform a floe-size distribution analysis on 78 high-resolution,
cloud-free, electro-optical satellite images of sea ice in the Canada Basin
acquired from a United States military passive satellite sensor from 1999
through 2014 (excepting years 2003–2005 and 2009), declassified as a part
of the military and scientific coalition MEDEA program (Baker and Zall,
2020), and distributed to the public through the United States Geological
Survey (USGS) Global Fiducials Library (GFL) Program. The images were
obtained during April through September of those years over various
geographic locations (Fig. 1a and Table 1), including three stationary
“fiducial” sites in the Beaufort and Chukchi Seas, and the northern Canada
Basin, designated as consistent locations within the Basin for inter-annual
comparison of environmental observations. The 2013 and 2014 image sets
contain additional images acquired at nonfiducial sites over designated
drifting floes and released through the GFL in support of the National
Aeronautics and Space Administration (NASA) Operation IceBridge, and the Office of Naval Research (ONR) Seasonal Ice Zone Reconnaissance Surveys (SIZRS) and Marginal
Ice Zone (MIZ) Departmental Research Initiative (DRI) field campaign (see
Lee et al., 2012). The images are panchromatic (with uncalibrated grayscale
pixel values ranging from 0 to 255) and projected onto the Universal
Transverse Mercator (UTM) grid with a resolution of 1 m; the SIZRS images
have a resolution of 1.3 m. The images cover areas O(1–1000) km2 and allow for characterization of the sea-ice FSD on
scales from O(1) m2 to O(100) km2. We note
that partially or fully cloud-covered images on the GFL were generally
unambiguous and rejected outright from our analysis. Cloudy pixels either
fully obscure information about the ice cover below or interfere with the
proper identification of floe outlines. For further descriptions of the MEDEA
imagery, see Kwok (2014) and Baker and Zall (2020).
Map of study region with image locations, and example subsets of
images and corresponding segmentations. (a) Study region within the Canada
Basin with locations of 78 images from 1999 to 2014 (gray circles). (b)
through (f) 100 km2 image subregions (top) and corresponding image
segmentations (bottom) from (b) 30 April 2014, (c) 12 June 2008, (d) 23 July 2007, (e) 11 August 2014, and (f) 20 September 2014. Locations of images (b) (green)–(f)
(yellow) are labeled on the map. USGS fiducial sites (black asterisks), for
which there are images from multiple years, are noted to the southeast of (e)
(Beaufort Sea), at (c) (Chukchi Sea), and at (d) (northern Canada Basin). Median
ice extents (bounding the area with more than 15 % concentration) are
shown for (b)–(f) in corresponding colored lines for those months (the April 2014
extent is south of the map domain). The median monthly ice extents are from
the NSIDC Sea Ice Index, Version 3 (Fetterer et al., 2017).
Number of images acquired at the Beaufort, Chukchi, and northern
Canada Basin fixed USGS fiducial sites (designated in Fig. 1a by asterisks);
at GFL locations corresponding to other programs (including NASA's Operation
IceBridge, and ONR's SIZRS and MIZ DRI, designated in Fig. 1a by light gray
circles at locations with no asterisks); and in total. Images acquired on
the same day at the same site are not independent samples; their presence is
denoted by *.
We examine the FSD for all 78 images in the context of the following
environmental parameters: sea ice concentration (SIC, fractional area of sea
ice in the image), distance to the ice edge (km), and surface air
temperature (SAT, ∘C), Table A1. SIC is calculated for each image
by dividing the total identified ice area (including that of
border-intersecting floes) in the segmented image by the total area of the
image. This is compared with SIC from passive microwave satellite data for
the dates and locations of the images. Distance to the ice edge is computed
as the distance (rounded to the nearest 100 km) between the image location
and the nearest point on the median ice edge contour (defined where the
concentration is 15 %) for the month and year of the image. SIC from
passive microwave data is from the National Oceanic and Atmospheric
Administration/National Snow and Ice Data Center (NOAA/NSIDC) Climate Data
Record of Passive Microwave Sea Ice Concentration, Version 4 (Peng et al.,
2013; Meier et al., 2021). Median ice edge contours are from the NSIDC Sea
Ice Index, Version 3, and are derived from passive microwave SIC data
(Fetterer et al., 2017). SAT (at 2 m) is retrieved from the European Centre
for Medium-Range Weather Forecasts (ECMWF) ERA5 Reanalysis (Hersbach et al.,
2020) hourly data on single levels from 1979 to present (Hersbach et al.,
2018) and taken as the mean daily value for each image region on the
corresponding image day.
Image segmentation
An algorithm for segmentation of individual sea-ice floes in the images is
developed (Denton, 2022) using a combination of “restricted growing”
steps (Soh et al., 1998), with the addition of an alternative approach
(described in Sect. 2.2.3) to the first step of the algorithm, which
requires the image to be preprocessed into a binary image. Generally, each
image is first manually classified into ice (floes) and water (background)
separated by some grayscale threshold based upon the image pixel value
histogram in which low grayscale values indicate dark open water, and high
values indicate bright ice. The classified image is then segmented via an
iterative erosion-expansion scheme in which floe-edge pixels are converted
to water pixels via a binary filter (see Soh et al., 1998) until a distinct
separation of individual floes is apparent (via visual check). The eroded
and separated floes are then individually labeled and subsequently expanded
to their original size (see Paget et al., 2001). Only the largest floes are
segmented and their ice pixels removed from the binary image after the first
erosion-expansion iteration, and the binary image is subsequently eroded
iteratively to lesser degrees to separate the remaining smaller, unsegmented
floes (see Stern et al., 2018b). Finally, any floes cut off by the image
borders are removed. Floe areas are retrieved from the segmented image to
construct an FSD, described in Sect. 2.3. We limit our FSD analysis to floes
having an area of at least 5 pixels, or 5 m2 (smaller scales
are indistinguishable from noise) and consider floe areas over the range of
5 m2 to 100 km2.
There are two main steps in erosion-expansion segmentation which require a
choice of parameter at the discretion of the user: classification and
erosion.
Classification: choice of grayscale threshold
Classification separates ice pixels from ocean pixels via the choice of a
threshold grayscale value. A grayscale optical satellite image of sea ice
ideally contains two peaks in its histogram: a bright-ice peak nearer to
values of 255 and a dark-ocean peak nearer to values of 0 (see Fig. 2g; note
that pixel values have been scaled to fall between 0 and 1). The threshold
must fall between the histogram ice and water peaks to separate ice floes
from the ocean. This choice of the precise threshold (see Sect. 2.2.3) can be
made difficult by the distance between the histogram peaks being large (as
in Fig. 2g), the peaks being flattened or nonexistent, or the presence of a
third peak or cluster of peaks between the ocean and bright-ice peaks,
resulting from classes that are not easily categorized as ice or ocean
(e.g., thin, dark ice, or melt ponds, or ridge shadows).
Comparison at the small-floe scale of segmentation of an MIZ MEDEA
image from 11 August 2014 using different parameters. (a) A 750 m × 750 m subregion of the image showing heavily ponded,
broken ice and open water; (b) through (f) segmentations of the same subregion
obtained by applying grayscale thresholds (on a scale of 0 to 1) of 0.15,
0.3, 0.4, 0.5, and 0.7, respectively; (g) histogram of pixel grayscale values
(scaled from 0–255 to 0–1) for the overall image showing the location of
grayscale thresholds for (b) (black) through (f) (magenta); and (h) floe size
distributions of (b) (black) through (f) (magenta). Vertical lines are shown at
scales shown in (b) through (d), which correspond to the size of artifact floes
in each.
Erosion and expansion
Erosion converts any ice pixels adjacent to ocean pixels in the classified
image into ocean pixels. This has the visual effect of eroding the ice floes
away from each other but also of expanding any clusters of ocean pixels in
floe interiors (e.g., melt ponds classified as ocean), possibly leading to
the division of a single floe into multiple floes. Erosion is done iteratively
enough times to provide a full separation of floes, with clear boundaries of
ocean between them. The eroded binary image is then filtered in a process
called filling, in which any ocean pixels in the interior of individual
floes are converted to ice pixels (see Stern et al., 2018b); this has the
visual effect of filling ocean holes in floes, and practically of suppressing
artifact floes from emerging in floe interiors during the subsequent
expansion step. The eroded, filled floes are then labeled with a unique
positive integer value.
The binary image is then filtered one last time in a process called
expansion, in which eroded pixel bands around floe edges (ocean pixels that
were originally ice pixels) are converted back to ice pixels, band by band,
the same number of times as the number of erosions. At every step, these
pixels are assigned a value equivalent to the positive integer mode of the
surrounding 8 pixels (or a new unique positive integer value if all
neighboring pixels represent ocean and have a value of 0), until all floes
are expanded to their original size with unique numerical labels (see Paget
et al., 2001). This process is repeated hierarchically in which the largest
floes are segmented and removed from the binary image first, and the
smallest floes are segmented last; this is because the number of erosions
required to separate the largest floes will also completely erode the
smallest floes, leaving them unlabeled in the segmented image (see Stern et
al., 2018b). The number of erosions at each hierarchical step is chosen such
that floe separation is maximized while expansion of ocean holes within
floes is minimized.
Selection of classification threshold
Past segmentation studies have chosen a classification threshold that
reduces features on the surface of a floe (e.g., melt ponds or ridge shadows
classified as ocean; see Paget et al., 2001, their Fig. 1a; and Stern et
al., 2018b, their Fig. 3b). The motivation for this choice, which is a lower
threshold value, is to avoid the growth of ocean holes in a floe and to
reduce artifact floes (see, e.g., Paget et al., 2001). On the other hand,
small floes (having horizontal scales less than around 40 m) do not separate
well after classification with a lower-threshold approach because the
grayscale pixel values of the boundaries of small floes tend to be similar
to that of surface features (see Fig. 2); small-floe boundaries will be
ill-defined in the classified image, and they may be assigned as belonging
to larger floes. Further, artifact floes emerge where small floes are
ill defined or where a few surface features have survived low-threshold
classification. These artifact floes are apparent by their rectangular edges,
which result from the row-by-row sweep across the image of the
erosion-expansion filters; if usually rounded floe edges or surface features
are not well defined, the effect of the filters will be to impose linear
edges and features.
FSDs and FSD slopes m versus month, SIC (fractional area) versus
month, and slopes m versus SIC, for 78 satellite images acquired from 1999 to
2014 in April through September of those years. (a) FSDs (gray lines) are
plotted on a log-log scale using 15 logarithmically spaced bins for the
range of floe areas spanning 5 m2 to 100 km2 and the requirement
of a bin count of at least 2 floes. A representative FSD is shown for the 19 June 2014 image (black line with solid circles) with a linear best-fit
(green dashed line) and slope m (green). Fits are taken from 50 m2 to 5 km2 for reasons discussed in Sects. 2.2.3 and 2.3. (b) FSD slopes m
versus month. In (b)–(d), black dots are shown for images segmented with
high confidence and gray dots for those segmented with low confidence.
Slopes m for the three GFL site (see Fig. 1a) images only are shown in cyan
(Beaufort Sea), red (Chukchi Sea), and magenta (northern Canada Basin), with
mean monthly slopes (triangles). (c) SIC versus month. SICs for the GFL site
images only are shown again in cyan (Beaufort Sea), red (Chukchi Sea), and
magenta (northern Canada Basin), with mean monthly SICs (triangles). In (b)
and (c), individual slopes and SICs from low-confidence segmentations at the
GFL sites are shown in a lighter shade of each site's designated fill-color.
(d) FSD slopes m versus SIC and linear fit (black dashed line) with slope (and
95 % confidence intervals).
To alleviate the issues described above, we take an alternative approach and
choose a threshold value that is sufficiently large that small-floe
(horizontal scales less than around 40 m) boundaries are well defined in the
binary image (Fig. 2e). Large floes remain well defined, even if the larger
threshold results in ocean pixels within their interiors. The number of
erosions required to properly separate floes at each hierarchical step is
much lower [O(1) compared to O(10); see e.g., Paget et al., 2001; Stern et
al., 2018b] if a larger threshold is chosen. Performing fewer erosions
limits the expansion of floe-interior ocean holes and the emergence of
artificial floes. The filling step also acts to alleviate emergence of
artifact floes. Examination of the histogram of pixel grayscale values for
any given image suggests a natural choice of threshold as the local minimum
between two local maxima (dark ice or melt ponds and bright ice). In practice,
the choice must usually be made using iterative adjustments to this location
after visual checks of the classified image; here, we iteratively increase
the threshold above the minimum until the edges of small floes are
appropriately delineated (see Fig. 2).
This choice of higher threshold yields adequate identification of smaller
floes in the image, with smaller and fewer artifact floes (those with
rectangular edges, see Fig. 2f compared with Fig. 2b). A secondary benefit
of the high-threshold approach presented here is speed. The expansion step
occupies the most time (mode filtering is a computationally intensive
process); because the number of expansions will match the number of
erosions, reducing the number of erosions by an order of magnitude will
significantly speed up the segmentation. In practice, we find that using the
low-threshold approach of Paget et al. (2001) on our dataset results in a
segmentation time of O(10) min to O(1) d, while using our
high-threshold approach results in a segmentation time of O(1) min to
O(1) h.
Floe size distribution
We construct the FSD using a number density n(a), computed as the fractional
number of floes in the scene having area between a and a+da, divided by the
width of the bin, da. We use 15 bins spaced logarithmically (such that bin
sizes increase with larger areas) from 5 m2 to 100 km2, with a
minimum floe number requirement of 2 per bin. If the FSD satisfies a
power law, the number density will fall along a straight line in a log-log
plot; we can write n(a)=cam for 0<a<∞, where c is a normalization
constant, and the distribution has slope m. We test the sensitivity of the
FSD to the choice of bin number by varying the number of bins from 15 to 5
and find that the shape of the FSD is stable between 10 and 15 bins. Due to
the sparsity of floes in the largest bins, a result of the finite size limit
of whole large floes being captured in satellite images, we limit the linear
fit in log-log plots (to estimate m) to floe areas smaller than 5 km2.
The FSD defined above is a noncumulative form, while some studies present
the cumulative form of the FSD (i.e., the integral of the probability
density function). When the noncumulative FSD is a straight line on a
log-log plot, its cumulative form will not be a straight line when the
maximum floe size has some finite upper bound. Rather, the cumulative FSD in
log-log space will be concave down (see the discussion by Stern et al.,
2018a). The cumulative FSD may present both a flattened slope over
small-floe scales and a steep slope in the large-floe tail (e.g., Hwang et
al., 2017, Fig. 1d), neither of which can be discerned as purely physical.
Interpretation of the cumulative FSD is ambiguous because this concave-down
behavior may alternatively be a manifestation of the distribution of ice
floe sizes having multiple power-law regimes.
The floe size may be taken to be floe area a, perimeter, or a diameter proxy
such as mean caliper diameter (MCD), used commonly after Rothrock and
Thorndike (1984). In the present work, we use floe area because we obtain
this directly in the segmentation (and it is directly relatable to floe
models), although this is easily related to the MCD (see Rothrock and
Thorndike, 1984; Stern et al., 2018a). We note, however, that relating FSDs
derived from a and MCD requires caution (see Sect. 3.4).
FSD sensitivity to the choice of classification threshold
The size of the “artifact” floes discussed in Sect. 2.2.3 (where the size
is shown in scale bars in Fig. 2b–d), corresponds to the scale of an
apparent change in slope of the corresponding FSD (Fig. 2h, black, red, and
blue dashed lines), in which the slope (exponent) is steeper for floe areas
larger than this scale and flatter for floe areas smaller. This appears to
result from an overidentification of floes at the artifact scale, and an
underidentification of the floes smaller than it. Testing a range of
classification thresholds shows how the scale of artifact floes is affected
by this choice, as is the resulting scale at which there is a change in FSD
slope (Fig. 2b through d); a higher threshold choice eliminates the spurious
change in FSD slope caused by such floe misidentifications around this
artifact scale.
A potentially undesirable effect of a high threshold is that larger floes
may be incorrectly divided into multiple floes (see Fig. 2c through f).
However, such oversegmentation of larger floes seems to have a minimal
effect on the slope of the FSD for floe areas larger than the artifact scale
(as in Fig. 2h). At the small-floe scale, the lower limit to the reduction of
the size of artifact floes through adequate segmentation is around 50 m2 for our image dataset. For this and the reasons described above, the
slope of the FSD is valid only for floe areas larger than 50 m2, and
we limit fitting in log-log plots (to estimate m) to floe areas between 50 m2 and 5 km2 (see also Sect. 2.3). The number of floes that fall
in this fitting range (see Table A1) is between 10 % to 55 % of the
total number of whole floes segmented across the entire floe range between 5 m2 and 100 km2 (with the minimum floe count requirement of 2 per
bin). Any segmentation resulting from this approach that identifies floes
inadequately or cannot be validated due to visual ambiguity of the ice field
is not included for analysis. Certain segmentations that, upon visual
validation, are neither wholly adequate nor inadequate are retained for
analysis but are tagged in plots in the results and are indicated in Table A1 as low confidence.
FSD power-law fit evaluation
The maximum likelihood estimator (MLE, see Clauset et al., 2009) can be
preferable for the determination of FSD slopes as it does not rely on
specifying bins or fitting ranges (Hwang et al., 2017; Stern et al., 2018a, b). In addition to least-squares fitted slopes m and
following Clauset et al. (2009), we compute FSD MLE slopes mMLE, and
conduct goodness-of-fit tests on these power-law fits, reporting
corresponding p-values (where the p-value is the probability that the difference
between the model fit and the observed FSD could be due to statistical
fluctuations; see Clauset et al., 2009 for a detailed discussion). The
power-law fit is a plausible model for the FSD if the computed p-value is
sufficiently large (p≥0.1); otherwise, the power-law model must be
rejected.
Clauset et al. (2009) argue that a strict statistical lower bound on
power-law behavior must be computed for the observed distribution; we
compute these values amin, following their methodology. Because
mMLE and amin are determined directly from the unbinned floe areas
for each image, we compute both over all floe areas (≥ 5 m2), and
do not exclude floes at or below the artifact scale.
ResultsFSD slope characteristics
Results indicate that FSDs are characterized by a single power law with
(linear least-squares fit) slope m for the entire regime of floe areas
between 50 m2 and 5 km2 (Fig. 3a). Slope values m range from -2.03
to -1.65 (Table A1) with a mean across all images of -1.79 ± 0.08. This
single power-law structure is consistent across all images (Fig. 3a), which
span 6 months from initial spring breakup in April to the September ice
minimum for a 15-year period, and a range of sea-ice settings from the
MIZ to the interior pack.
We find no significant difference between slopes m and mMLE (Table A1).
The mean mMLE over all images is -1.77 ± 0.11. Considering each
image, mMLE differs from m by about 3 % on average. We find that 76 %
of the fits pass the goodness-of-fit test with p≥0.1 (Table A1),
meaning that the FSDs can plausibly be power-law distributed. Finally, we
find that the strict lower-bound to power-law behavior amin varies
considerably over the images (Table A1), spanning around 10 to 10 000 m2 with a median value of 361 m2. Considering that the largest
floe areas in the images are around 10 to 100 km2, the range of floe
sizes over which the power-law fits apply is large. Values amin can vary
significantly even across images acquired on the same day at the same
location (see, e.g., Table A1, images 14–15).
Examining m from 1999 to 2014 reveals that there is no apparent overall
interannual trend of FSD slopes in the Canada Basin. It might have been
expected that a steepening of the slope (i.e., a higher portion of small to
large floes, and more negative FSD slopes) over multiple years would occur
as the sea-ice thins and summer concentrations decline. However, we find no
evidence for an overall change in m. It may be that the latitudinal span of
images obscures any temporal variability over the 15 years analyzed.
Partitioning the image FSD slopes by month, we find that there is no
apparent variation in m with season (Fig. 3b). In the next section, we
consider FSD slopes retrieved at the three fixed GFL fiducial sites (see
Fig. 1a) to investigate whether there may be a seasonal signal obscured by
spatial variability of the sample locations. There is an increasing spread
in the values of m in any single month as the season progresses from April
through September. We will show that in these later months, the broad
latitudinal distribution in images is accompanied by a significant
latitudinal distribution of SICs and SATs. We further note that we have low
confidence in some segmentations in late summer months (those shown by gray
dots in Fig. 3b–d); appropriate segmentation of images in which the effects
of melt are prominent (e.g., extensive ponding and slush ice) can be
problematic, especially when validation by eye is not possible.
Seasonal variability at stationary locations in the Canada Basin
Considering only the 17 images at the Beaufort Sea site (73∘ N,
150∘ W), which span the whole range of years and months, we find
that a clear seasonal signal in slope emerges (Fig. 3b, cyan line). The mean
slope m at the Beaufort site is shallowest in April and May, and then
steepens through August, increasing only slightly through September (only a
single image is available for each month from July to September at the
Beaufort site).
At the Chukchi site (70∘ N, 170∘ W), 10 images span years
2006–2014 and only for months April through July. While we cannot examine
the entire spring–fall seasonality of Chukchi FSD slopes, there is evidence
of a similar start to the seasonal signal as that of the Beaufort, with
greater variability in April and June. In April and May, mean m at the Chukchi
site is shallowest, steepening for June and July (Fig. 3b, red line).
Examining m from 16 images spanning years 2000–2014 and the entire range of
months at the northern Canada Basin site (85∘ N, 120∘ W), we
find no discernable seasonal variability of FSD slopes (Fig. 3b, magenta
line). We posit that a lack of seasonal signal in the northern Canada Basin
is due to the lack of a seasonal signal in SIC at that location, which we
discuss in the next section.
Note that for each of the GFL sites, we compute mean monthly slopes after
first taking the mean of any slopes from images acquired on the same day at
a particular site, to account for the fact that these are not independent
estimates (see Table 1). We do the same for mean monthly SIC at the GFL
sites, discussed in the next section.
Relationship between FSD slope, sea-ice concentration, and surface air
temperatures
It is notable that seasonal variations in SIC are only apparent for the
images at the Beaufort and Chukchi sites, and not at the northern Canada
Basin site (Fig. 3c). At both the Beaufort and Chukchi sites, the evolution
of monthly mean SIC (highest in April and May and decreasing through the
summer) closely resembles the seasonal evolution of m for the sites. Mean SIC
at the northern Canada Basin site exhibits virtually no seasonality and
always remains above 0.80, in the same way that m does not vary much from
spring to fall at that site.
There is a statistically significant linear relationship between m and SIC
(Fig. 3d), with m shoaling as SIC increases. The best-fit linear slope is the
same (within 95 % uncertainty) if values of m for segmentations with poor
confidence (gray dots) are excluded from the fit. Note that there are more
sample points in the high SIC range than in the low range, and the linear
fit can only explain 33 % of the variation (r-squared is 0.33) in m with
SIC. However, the linear relationship is statistically significant with a
p-value of O(10-8) (i.e., < 0.01).
The FSD may logically be expected to differ with distance to the ice edge
if, for example, wave propagation into the ice pack plays some role in
governing floe breakup (see the discussion in Toyota et al., 2011, 2016). In the set of images analyzed here, the variation in m with
distance to the ice edge (not shown) is not straightforward. For those
images with SIC less than 0.80, which range from 0 to 1600 km from the ice
edge, m appears to generally shoal with increasing distance from the ice edge.
However, for images with SIC greater than or equal to 0.80, which range from
200 to 3600 km to the ice edge, m exhibits no clear variation with distance
to the ice edge. SIC is not linear with distance to the ice edge at the
location of images analyzed here; any tie between distance to the ice edge
and m is likely dominated by SIC.
With respect to SAT, we find that m is relatively constant (between around
-1.9 to -1.7) for a large range of temperatures (mean SAT over the day of a
given image), in the range -25 to -2 ∘C, with no statistically
significant linear relationship between m and SAT. In a “melt” regime
(which we define to correspond to SATs between -2 and 4 ∘C), m values
span their entire range (between around -2.0 and -1.6). This shows, again, the
increased range of FSD slopes during the warmer months. Considering only the
Beaufort and Chukchi Sea GFL sites reveals a similar structure to the two
overall temperature regimes for FSD slopes: a cold regime in which values of
m remain relatively constant, and a melt regime in which values of m span
nearly their entire range. At the northern Canada Basin site, on the other
hand, SATs remain predominately below 0 ∘C, and m remains shallow in
the melt regime between temperatures -2 and 0 ∘C (i.e., there are no
m values <-1.82). Finally, we note that for this image set, there is
no clear relationship between SIC and SAT, again because for the melt range
of SATs, SICs span their entire range. That is, the SIC relates directly to
the FSD slope m, while there is no relationship between SAT and m.
Context with previous studies
It is useful to compare our slopes to relevant previous studies in the same
region (Stern et al., 2018b, a; Hwang et al., 2017). Stern
et al. (2018b) examined the noncumulative FSD using MCD, x, and plotting a
floe number density n(x), where a∼x2. From an application of basic
probability theory, slopes reported in studies that examine the
noncumulative FSD using normalized floe number densities constructed from
x are equivalent to 2m+1 (where m refers to slopes found in this study). Note
that this is not the same for the comparison to slopes of the cumulative FSD
(see Stern et al., 2018a, their Table 1 footnotes).
Stern et al. (2018b) analyzed moderate-resolution (250 m) satellite images
and characterized the FSD in the Beaufort and Chukchi seas during the
summers of 2013 and 2014, finding that a single power law describes the FSD
across floe diameters 2 to 30 km. Applying the transformation above to the
reported range of slopes in Stern et al. (2018b) (-2.81 to -1.90, their
Table 4) yields -1.91 to -1.45, which overlaps closely with the range of m
found here. We do not expect complete overlap of our slope range with theirs
as they report mean monthly slope values, whereas our range is reported for
the entirety of segmented images. We note that for our analysis of the same
subset of an image analyzed by Stern et al. (2018b) (8 July 2014, their Fig. 10; image not included in our analysis due to partial cloud cover), our
segmentation characterized by FSD slope m agrees exactly with theirs (upon
applying the transformation).
Stern et al. (2018b) additionally analyzed the FSD in 12 subregions of 3
high-resolution MEDEA images in 2014 in the Beaufort Sea and concluded that
a single power-law characterization may extend to floe scales as small as 10 m, although the authors note that this conclusion is only supported by
visual comparison of the FSD slopes on the smaller scale and those on the
larger scale (from the moderate-resolution images), and not from
statistical, quantitative comparison. Here, we extend the study of the
small-scale behavior of the FSD from 3 high-resolution images over 1
summer to 78 over 12 summers in the same Arctic region and surrounding
it, and find that a single power law is, indeed, applicable to the FSD across
floe areas of 50 m2 to 5 km2, equivalent to a floe diameter range
of ∼ 9 m to 3 km (using the area to MCD relation in Rothrock
and Thorndike, 1984, a=0.66x2).
With respect to seasonal variability, Stern et al. (2018b) found similar
seasonal variations for floe sizes in the 2 to 30 km range in Beaufort and
Chukchi FSD slopes (steepening from April through August and shoaling again
in September). They point out that this is consistent with spring through
summer breakup of larger floes, the shrinking of floes due to summer melt,
followed by removal of the smallest floes at the end of melt, and fall
freeze-up of the ice field into large floes again. Hwang et al. (2017)
examined the cumulative FSD for floe MCDs larger than about 100 m using
TerraSAR-X Synthetic Aperture Radar images from 2014 in the Beaufort Sea
region. They relate floe fracturing and corresponding steepening in FSD
slopes (over a similar range of scales described by m) to a sequence of
wind-driven deformation events over one summer season in the Beaufort Sea.
They demonstrate a distinct steepening of the FSD slope in August, which they
relate to the timing of melt becoming dominant.
Summary and Discussion
We have segmented and retrieved the areas of Arctic sea-ice floes from 78
high-resolution optical satellite images acquired in the Canada Basin
between 1999 and 2014. Our analysis of the resulting FSDs shows that the
distributions exhibit a single power-law behavior across floes ranging in
area from 50 m2 to 5 km2. We find that the slope m of the power law
in log-log space ranges from -2.03 to -1.65 and shoals with increasing SIC.
We find that, correspondingly, at locations within the Canada Basin that
experience a distinct reduction in SIC from April through August and an
increase in September, a similar seasonal signal in m appears. On the other
hand, at locations that undergo no distinct change in SIC through the
summer, m remains constant.
While we might have anticipated that any seasonality in m might be related to
seasonal changes in SAT, consistent with melt onset (see, e.g., Hwang et al.,
2017; Stern et al., 2018b), we find that seasonal variation in m is more
directly related to changes in SIC. These findings provide support for an
approach that uses SIC in any characterization of the FSD. Future studies
are needed to investigate the relevant dynamics (i.e., wind-forced sea-ice
deformation and breakup) and thermodynamics (e.g., ocean-to-ice heat fluxes)
of the sea-ice pack to explore the precise mechanisms by which the sea-ice
concentration relates to the structure of the FSD, and how this relationship
might differ in different settings. For example, a scenario might be
envisioned where the FSD slope could steepen (e.g., as a result of fewer
large floes and more smaller floes), while the SIC remains the same, and
this might indicate a fracturing. Conversely, a shoaling of the FSD slope
associated with the loss of small floes (e.g., via relatively rapid lateral
melt of smaller floes compared to larger floes) may be associated with a
different SIC-FSD relationship.
Finally, we point out that several other previous studies report two
distinct floe-size regimes, in which a small-floe regime is characterized by
shallower FSD slopes and a large-floe regime by steeper slopes (Geise et
al., 2017; Steer et al., 2008; Toyota et al., 2011, 2006).
Using images from the Weddell Sea, Steer et al. (2008) examine the
non-cumulative FSD for floe diameters between O(1) and O(100) m, finding a
change in FSD slope at 20 m. In addition, Perovich and Jones (2014) show a
possible plateauing of the FSD slope at the small-floe scale (although they
do not explicitly refer to two regimes). For a range of sea-ice settings,
and considering floe diameters in the range O(1–1000) m, Toyota et al. (2011, 2006) find two floe-size regimes for floe sizes
larger and smaller than about 20 to 40 m diameter. We note that these
studies classify images into ice and water as an initial step, choosing a
classification threshold. Our test of FSD sensitivity to this choice reveals
that the FSD can appear divided into two power-law regimes if this choice
does not adequately identify small floes.
Our finding of a single power law suggests that the processes that govern
the distribution of floe sizes are similar across the full range of floe
sizes, while studies that find two distinct power-law regimes would
indicate that different processes act on different scales. For example,
Horvat and Tziperman (2017) use a coupled ice-ocean model to show that
increased lateral melt on specific floe sizes and transient oceanic forcing
on the ice pack can perturb the FSD behavior from a single power-law at the
relevant scale. Future work is needed to determine how different FSD
structures might emerge in certain settings.
Image number with corresponding GFL site (where B = Beaufort
Sea, C = Chukchi Sea, and NCB = northern Canada Basin are the fixed
sites; and IB = IceBridge, SIZRS = Seasonal Ice Zone Reconnaissance
Surveys, and MIZ = Marginal Ice Zone are corresponding program locations),
date, latitude and longitude (decimal degrees), image area (total area
viewed by the sensor, rounded to the nearest km2), total ice area
(combined area of all ice identified in the image including that of
border-intersecting floes, rounded to the nearest km2), sea ice
concentration (fractional area, SIC, where the first value is derived from
the image and the second, in parentheses, from the NOAA/NSIDC Climate Data
Record, CDR), approximate distance to the median ice edge of that month
(rounded to the nearest hundred km), SAT (2 m mean daily,
∘C), total number of whole floes retrieved (between 5 m2 and 100 km2 with the requirement of 2 per bin), total
whole-floe area (combined area of all whole floes identified in the image
after clearing image border-intersecting floes, rounded to the nearest
km2), number of floes with areas in the slope-fitting range (between 50 m2 and 5 km2), FSD linear best-fit slope m± slope
fit error e (95 % confidence), FSD maximum likelihood estimate slope
mMLE± slope fit uncertainty eMLE (standard
deviation over iterative fits), strict lower bound on MLE-fitted power-law
behavior amin, p-value on the MLE-fitted power-law model, and segmentation
confidence (where L = Low and H = High). Images acquired on the same day at the same site are denoted by *. SIC is calculated by
dividing the total ice area by the image area; note that this calculation is
performed prior to rounding the areas displayed in this table. SIC CDR is
from the NOAA/NSIDC Climate Data Record of Passive Microwave Sea Ice
Concentration, Version 4 (Peng et al., 2013; Meier et al., 2021). Median ice
edge contours are from the NSIDC Sea Ice Index, Version 3 (Fetterer et al.,
2017). SAT data are from the ECMWF ERA5 Reanalysis hourly data on single
levels from 1979 to present (Hersbach et al., 2018).
The MATLAB algorithm (Denton, 2022) written and used here to segment sea-ice floes in satellite images is available at
10.5281/zenodo.6146144. The sea-ice floe segmentation
products (Denton and Timmermans, 2022) derived from MEDEA imagery and
presented here are available for download at
10.5281/zenodo.6341621. MEDEA images are available from the
USGS GFL
(https://www.usgs.gov/global-fiducials-library-data-access-portal, United States Geological Survey, 2022).
SIC passive microwave data are from the NOAA/NSIDC Climate Data Record of
Passive Microwave Sea Ice Concentration, Version 4 (Peng et al., 2013; Meier
et al., 2021). Median ice edge contours are from the NSIDC Sea Ice Index,
Version 3 (Fetterer et al., 2017). SAT data are from the ECMWF ERA5
Reanalysis (Hersbach et al., 2020), ERA5 hourly data on single levels from
1979 to present (Hersbach et al., 2018), and were downloaded from the
Copernicus Climate Change Service (C3S) Climate Data Store. The results
contain modified Copernicus Climate Change Service information 2021. Neither
the European Commission nor ECMWF is responsible for any use that may be
made of the Copernicus information or data it contains.
Author contributions
AAD formulated the image segmentation algorithm, performed the analysis,
and took the lead in writing the article. Both authors shaped the
research, analysis, and writing.
Competing interests
The contact author has declared that neither they nor their co-author has any competing interests.
Disclaimer
Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Acknowledgements
The authors would like to thank Hans C. Graber of The University of Miami
Center for Southeastern Tropical Advanced Remote Sensing for providing
several of the MEDEA images used in this study.
Values mMLE, eMLE, amin, and the corresponding p-values were
computed directly using Aaron Clauset's MATLAB power-law fitting package,
available at https://aaronclauset.github.io/powerlaws/, last access: 22 April 2022). We thank the
editor, Chris Horvat, and one anonymous reviewer for their valuable comments
and suggestions.
Financial support
This research has been supported by the Office of Naval Research under the Multidisciplinary University Research Initiative on Mathematics and Data Science for Improved Physical Modeling and Prediction of Sea Ice (grant no. F1168-01).
Review statement
This paper was edited by Stephen Howell and reviewed by Christopher Horvat and one anonymous referee.
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